2. Inscribed Angle:An inscribed angle is an angle with its vertex "on" the circle, formed by two
intersecting chords.

Inscribed Angle =
Intercepted
Arc

<ABC is an inscribed angle.
Its
intercepted arc is the minor arc from A to C.m<ABC = 50°

Special situations
involving inscribed angles:

An angle inscribed in a
semi-circle is a right angle.

In a circle, inscribed
circles that intercept the same arc are
congruent.

A quadrilateral
inscribed in a circle is called a cyclic
quadrilateral.

The opposite angles in
a cyclic quadrilateral are supplementary.

3. Tangent Chord Angle:An angle formed by an intersecting tangent and chord has its vertex "on" the circle.

Tangent ChordAngle =Intercepted
Arc

<ABC is an angle formed by a tangent and chord.
Its intercepted
arc is the minor arc from A to B.m<ABC = 60°

4. Angle Formed Inside of a
Circle by Two
Intersecting Chords:When two chords intersect "inside" a circle, four angles
are formed. At the point of intersection, two sets of vertical
angles can be seen in the corners of the X that is formed on the
picture. Remember: vertical angles are equal.

Angle
Formed Inside by Two
Chords =Sum
of Intercepted Arcs

Once you have found ONE of these angles, you
automatically know the sizes of the other three by using your
knowledge of vertical angles (being congruent) and adjacent angles forming
a straight line (measures adding to 180).

<BED is formed by two intersecting chords.
Its
intercepted arcs are
.
[Note: the intercepted arcs belong to the set of vertical
angles.]
also, m<CEA = 120° (vetical angle)m<BEC and m<DEA = 60° by straight line.

5. Angle Formed Outside of a
Circle by the Intersection of:"Two Tangents" or "Two Secants"
or "a Tangent and a Secant".

The formulas for all THREE of these
situations are the same:
Angle Formed Outside = Difference
of Intercepted Arcs (When subtracting, start with the larger
arc.)

Two Tangents:<ABC is formed by two tangents
intersecting outside of circle O.
The intercepted arcs are minor arc
and major arc
.
These two arcs together comprise the entire circle.

Special situation for
this set up: It can be proven that <ABC
and
central <AOC are supplementary. Thus the angle formed by
the two tangents and its first intercepted arc also add to 180º.

Two Secants:<ACE is formed by two secants
intersecting outside of circle O.
The intercepted arcs are minor arcs
and
.

a Tangent and a Secant:<ABD is formed by a tangent and a secant
intersecting outside of circle O.
The intercepted arcs are minor arcs
and
.